If a space curve $\gamma(s)$ parametrised with respect to arc length $ s $ has curvature as $$\kappa(s) = \cos^{2} (\theta(s))$$ and torsion as $$\tau(s) = \sin(\theta(s) ) \cos(\theta(s)) ,$$ where $ \theta(s) $ is a function that gives the angle between the tangent vector to the curve and it's Darboux vector i.e. $ \tau T + \kappa B $ , then is there a way of getting the possible parametrisation of such a curve or even $\textbf{an example of such a curve}$??

Of course, apart from the circular helix, in which case the function $\theta $ is a constant.

$\textbf{Note}$ : The angle $\theta(s)$ is actually the angle between the tangent to $\gamma$ and the tangent vector of another curve which is $1-1$ correspondence with $\gamma$, however that tangent vector is itself parallel to the Darboux vector, so I have defined it as such.

  • $\begingroup$ Do you mean a curve for which $\kappa$ and $\tau$ satisfy the given conditions? If so, you shouldn't expect many solutions: We can write $\cos^2 \theta$ and $\sin \theta \cos \theta$ as functions of $\kappa$ and $\theta$, giving a system of two equations in two unknowns, which generically has finite many solutions. In that case, $\kappa$ and $\tau$ must be locally constant, so one expects the only solutions to be certain helices. $\endgroup$ – Travis Jun 9 '17 at 12:36
  • $\begingroup$ @Travis Yes. I did mean the same. But I didnt get why they would be helices as that would force $\theta$ to be a constant, right?? By locally constant, did you mean, like a curve patched together by helices of different radius? $\endgroup$ – Vishesh Jun 9 '17 at 16:33
  • $\begingroup$ So, if one carries out the computation I described, you get the solutions $\kappa = \frac{1}{2}$, $\tau = \pm \frac{1}{2}$. But the only curves with constant curvature and torsion are helices. In fact, that this is the solution to the system shows that there's only a single solution up to Euclidean motions (allowing reflections). $\endgroup$ – Travis Jun 9 '17 at 16:36
  • $\begingroup$ @Travis. Well the curvature and torsion functions seem to satisfy a relation like $\kappa^2 +\tau^2 = \kappa$ and I could think of non-helical curves too which satisfy that relation. So I presumed there may be other curves than the helix. Also I didn't get as to how you ended with those values. The varying angle condition seems to be vacuously satisfied then. But the fundamental theorem seems to hint that s not the case, right? $\endgroup$ – Vishesh Jun 9 '17 at 16:59

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